## Tag `0062`

Chapter 5: Topology > Section 5.19: Specialization

Lemma 5.19.2. Let $X$ be a topological space.

- Any closed subset of $X$ is stable under specialization.
- Any open subset of $X$ is stable under generalization.
- A subset $T \subset X$ is stable under specialization if and only if the complement $T^c$ is stable under generalization.

Proof.Omitted. $\square$

The code snippet corresponding to this tag is a part of the file `topology.tex` and is located in lines 3230–3240 (see updates for more information).

```
\begin{lemma}
\label{lemma-open-closed-specialization}
Let $X$ be a topological space.
\begin{enumerate}
\item Any closed subset of $X$ is stable under specialization.
\item Any open subset of $X$ is stable under generalization.
\item A subset $T \subset X$ is stable under specialization
if and only if
the complement $T^c$ is stable under generalization.
\end{enumerate}
\end{lemma}
\begin{proof}
Omitted.
\end{proof}
```

## Comments (0)

## Add a comment on tag `0062`

Your email address will not be published. Required fields are marked.

In your comment you can use Markdown and LaTeX style mathematics (enclose it like `$\pi$`

). A preview option is available if you wish to see how it works out (just click on the eye in the lower-right corner).

All contributions are licensed under the GNU Free Documentation License.

There are no comments yet for this tag.